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Stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises
Journal of Sound and Vibration 333 (2014) 954–961
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Stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises H.T. Zhu n State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, People's Republic of China
a r t i c l e in f o
abstract
Article history: Received 26 August 2013 Accepted 2 October 2013 Handling Editor: K. Worden Available online 1 November 2013
This study presents a solution procedure for the stationary probability density function (PDF) of the response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises. First the Zhuravlev non-smooth coordinate transformation is adopted to convert a vibro-impact oscillator into an oscillator without barriers. The stationary PDF of the converted oscillator is governed by the Fokker–Planck (FP) equation. The FP equation is solved by the exponential-polynomial closure (EPC) method. Illustrative examples are presented with vibro-impact Duffing oscillators under external and parametric Gaussian white noises to show the effectiveness of the solution procedure. The parametric excitation is acting in displacement and the constraint is a unilateral zerooffset barrier. The restitution coefficient of impacts is taken as 0.90. Comparison with the simulated results shows that the proposed solution procedure can provide good approximate PDFs for displacement and velocity although a little difference exists in the tail of these PDFs. This difference may be due to the weak approximation on the response of the vibro-impact oscillators using a continuous Markov process when the restitution coefficient is not very close to unity. & 2013 Elsevier Ltd. All rights reserved.
1. Introduction The stochastic response of structures or systems under random excitation is a critical issue in reliability analysis. An overlarge response may be a great threat to the safety of life and property. Therefore, many efforts are made on the evaluation of the stochastic response of these systems. In particular, vibro-impact systems have attracted much attention in the scientific community because the systems can model many relevant problems in the field of science and engineering. For example, vibro-impact systems can describe ship roll motion against icebergs, rotor–stator rubbing in rotating machinery or impact interactions in pipe–baffle interfaces due to seismic excitations, etc. The modeling, mapping and applications of vibro-impact systems are extensively introduced by Ibrahim [1]. The behavior of vibro-impact systems has been widely studied in the case of deterministic or random excitations. In the case of deterministic excitation, many interesting dynamic phenomena were observed, e.g., subharmonic oscillation [2], impact modes [3], chaotic motion [4,5], grazing bifurcation [6–8]. In the case of random excitation, many researches have also been conducted and reported. Dimentberg and Iourtchenko presented a comprehensive review on the investigation on vibro-impact systems under stochastic excitations [9]. In those investigations, the stochastic response of vibro-impact systems was evaluated by statistical moments or probability density functions [10]. An equivalent linearization method was used for analyzing
n
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0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.10.002
H.T. Zhu / Journal of Sound and Vibration 333 (2014) 954–961
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random vibration of a beam impacting stops [11]. This method is limited to the case of light nonlinearity and nearly elastic impact. A stochastic averaging method is another approach to deal with the problem of vibro-impact systems [12–15]. The stochastic averaging method is suitable in the case of lightly damping and weak wide-band excitation. Monte Carlo simulation is a versatile technique [16]. However, extensive efforts are needed to simulate impacts. The procedure of impact tracing also introduces additional points when simulating each impact. So, in order to simulate an adequate PDF, both response interval selection and sample selection around the barrier are critical in this simulation technique. Besides, when the tail of response PDF is considered, the simulation time becomes tremendous. On the other hand, the Duffing oscillator in its various forms has been adopted to model many nonlinear physical systems, such as large deflection of a beam with nonlinear stiffness, vibration of a beam with nonlinear stiffness due to inplane tension, nonlinear cable vibrations and nonlinear electrical circuit [17]. Furthermore, consideration of Duffing-type nonlinearities in some mechanical systems can lead these mechanical systems to improve their performance. For example, introduction of Duffing-type nonlinearities into a mono-stable energy harvester can reduce the maximum displacement amplitude of the center magnet without effecting device performance, which can be used to create smaller energy harvesting devices [18]. In this research field, most previous researches were devoted to the case of sinusoidal excitation. The case of ambient vibration has been becoming a popular research topic recently because an energy harvesting device sometimes performs ambient vibration which is stochastic in nature [19]. When ambient vibration is considered, the excitation is usually treated as Gaussian white noise and the response of this device can be evaluated using the Fokker– Planck (FP) equation. Therefore, an investigation on the Duffing oscillator in its various forms under different types of excitations is fundamental for further investigating many nonlinear physical systems. This paper presents a study on the PDF of the stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises. The constraint is a unilateral zero-offset barrier. The vibro-impact Duffing oscillator can be used to model random vibration of a beam with a unilateral barrier under axial and transversal random excitations [20,21]. In particular, the probability density function (PDF) and the tail PDF of these Duffing oscillators are formulated, which is less addressed in previous studies. In this solution procedure, the Zhuravlev non-smooth coordinate transformation is first introduced to the equation of motion so that the transformed equation does not contain any impact terms [1,22]. The PDF of the response of the transformed oscillator is governed by the FP equation which is solved by the exponentialpolynomial closure (EPC) method [23–26]. The cases of light nonlinearity and strong nonlinearity are further investigated as well as the intensity level of parametric excitation. In these analyses, the restitution coefficient is 0.90, which is not very close to unity. Comparison with the simulated results shows that the proposed solution procedure can provide good approximate PDFs for displacement and velocity although a little difference exists in the tail of these PDFs. The formulation of this difference may be due to weakly modeling the response of the vibro-impact oscillators using a continuous Markov process when the restitution coefficient is not very close to unity. 2. Problem formulation A vibro-impact Duffing oscillator under a unilateral zero-offset barrier can be formulated by y€ þcy_ þ ky þ μy3 ¼ ξ1 ðtÞ þɛyξ2 ðtÞ; y_ þ ¼ r y_ ;
y 40
y ¼ 0; 0 o r r1
(1) (2)
€ y; _ y are the acceleration, velocity and displacement of the oscillator, respectively; c is the damping coefficient; k is where y; the linear stiffness coefficient; μ is the nonlinearity coefficient and ξi ðtÞ are the zero-mean mutually independent Gaussian white noises. The vibro-impact Duffing oscillator can be used to model random vibration of a beam with a unilateral barrier under axial and transversal random excitations [20,21]. The mean and correlation function of ξi ðtÞ are given below E½ξi ðtÞ ¼ 0
(3)
E½ξi ðtÞξi ðt þ τÞ ¼ 2πK i δðτÞ
(4)
where E½ is the expectation; 2πK i is the excitation intensity of ξi ðtÞ and δðÞ is the Dirac delta function. The use of the Zhuravlev non-smooth coordinate transformation leads the above vibro-impact oscillator to be converted into an oscillator without barriers. The detailed procedure was presented in either the monograph of Ibrahim [1] or the review paper written by Dimentberg and Iourtchenko [9]. The transformation procedure is briefly presented herein y ¼ jzj;
y_ ¼ z_ sgnðzÞ;
y€ ¼ z€ sgnðzÞ
(5)
where z€ ; z_ ; z are the acceleration, velocity and displacement of the converted oscillator, respectively, and sgnðÞ is the sign function. Eq. (5) is formulated because y ¼ jzj is equivalent to y ¼ z sgnðzÞ and z½dðsgnðzÞÞ=dt ¼ 0 noting that sgn(z) changes its sign at z¼0. Eq. (2) gives the condition for the case of inelastic impacts. In the transformation procedure, the coefficient r is assumed to be close to unity. Therefore, (1 r) can be regarded as a small parameter. In such a case, the response of nonlinear systems may be much less significant discontinuous in its time-derivative. This allows some conventional approximation techniques
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solving nonlinear stochastic differential equations to be used [10,14]. For example, the response of vibro-impact oscillators can be approximated with a continuous Markov process. Consequently the PDF of the response can be obtained by solving the corresponding FP equation. According to Eq. (5), Eq. (2) is converted to z_ þ ¼ r z_
at z ¼ 0
(6)
where z_ ; z_ þ represent the impact velocities before and after an impact for the converted oscillator, respectively. Therefore, the converted velocity jump is reduced by an amount proportional to (1 r). It is reasonable to use the Dirac delta-function to introduce this jump into the equation of motion as an additional impulsive damping term. The additional damping term due to impacts may be written in the form: ðz_ þ z_ Þδðt t i Þ ¼ ð1 rÞz_ δðt t i Þ;
given that jz_ þ j o jz_ jo jz_ j
(7)
where ti is the time instant of impacts (i.e., zðt i Þ ¼ 0). Correspondingly, the variables are converted from the time domain to the space domain. This can be fulfilled by setting ðt t i Þ ¼ zðtÞ=z_ ðt i Þ. It is because zðtÞ ¼ zðt i Þ þ z_ ðt i Þðt t i Þ in a small interval and zðt i Þ ¼ 0 at an impact. Therefore, δðt t i Þ ¼ δðzðtÞ=z_ ðt i ÞÞ and δðt t i Þ ¼ jz_ jδðzÞ. In this case, Eq. (8) can be written in the form: ð1 rÞz_ δðt t i Þ ¼ ð1 rÞz_ jz_ jδðzÞ
(8)
After the transformation process, Eqs. (1) and (2) are combined into one equation by adding the above impulsive damping term: z€ þ cz_ þ kzþμz3 þ ð1 rÞz_ jz_ jδðzÞ ¼ sgnðzÞξ1 ðtÞ þɛzξ2 ðtÞ Let x1 ¼ z, x2 ¼ z_ , Eq. (9) can be expressed as ( x_ 1 ¼ x2 x_ 2 ¼ cx2 kx1 μx31 ð1 rÞx2 jx2 jδðx1 Þ þ sgnðx1 Þξ1 ðtÞ þ ɛx1 ξ2 ðtÞ
(9)
(10)
The response vector fx1 ; x2 gT is Markovian and the PDFs of responses are governed by the following FP equation: ∂p ∂p ∂ ¼ x2 þ cx2 þkx1 þ μx31 þ ð1 r Þx2 x2 δðx1 Þ p ∂t ∂x1 ∂x2 þ
1 ∂2 p 1 ∂2 p 2πK 1 ðsgnðx1 ÞÞ2 2 þ 2πK 2 ɛ2 x21 2 2 ∂x2 2 ∂x2
(11)
In this paper, the stationary PDF solution is considered and the term on the left side of Eq. (11) is zero. Therefore, Eq. (11) is reduced as follows: x2
∂p þ c þ ð1 r Þx2 δðx1 Þ þ ð1 r Þx2 sgnðx2 Þδðx1 Þ p ∂x1 ∂p þ cx2 þkx1 þ μx31 þ ð1 r Þx2 jx2 jδðx1 Þ ∂x2 þ πK 1
∂2 p ∂2 p þ πK 2 ɛ 2 x21 2 ¼ 0 2 ∂x2 ∂x2
(12)
Considering jx2 j ¼ x2 sgnðx2 Þ, Eq. (12) finally reads x2
∂p þ c þ 2ð1 r Þx2 δðx1 Þ p ∂x1
∂p þ cx2 þkx1 þ μx31 þ ð1 r Þx2 jx2 jδðx1 Þ ∂x2 þ πK 1
∂2 p ∂2 p þ πK 2 ɛ 2 x21 2 ¼ 0 2 ∂x2 ∂x2
(13)
In general the following requirements are satisfied by the PDF pðx1 ; x2 Þ of the stationary response of the nonlinear oscillator: 8 > pðx1 ; x2 Þ Z 0; x1 ; x2 A R2 > > < lim pðx1 ; x2 Þ ¼ 0; i ¼ 1; 2 (14) xi - 7 1 > > R > þ1 R þ1 : 1 1 pðx1 ; x2 Þ dx1 dx2 ¼ 1 where R denotes real space. The exact PDF solution of Eq. (13) is usually not obtainable. According to the above requirements, an approximate PDF is proposed and it is expressed as an exponential-polynomial function of state variables. ~ 1 ; x2 ; aÞ to Eq. (13) is assumed to be [23–26] The approximate PDF solution pðx ~ 1 ; x2 ; aÞ ¼ C eQ n ðx1 ;x2 ;aÞ pðx
(15)
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where C is a normalization constant and a is an unknown parameter vector containing Np entries. The polynomial Q n ðx1 ; x2 ; aÞ is expressed as n
i
Q n ðx1 ; x2 ; aÞ ¼ ∑ ∑ aij xi1 j xj2
(16)
i¼1j¼0
which is an nth-degree polynomial in x1 and x2. To fulfill the requirements (14), it is also required that lim Q n ðx1 ; x2 ; aÞ ¼ 1;
i ¼ 1; 2:
xi - 7 1
(17)
~ 1 ; x2 ; aÞ is only an approximation of pðx1 ; x2 Þ and the number of unknown parameters Np is always limited in In general, pðx ~ 1 ; x2 ; aÞ practice. Therefore, the FP equation (13) cannot be satisfied exactly with the approximate solution. Substituting pðx into Eq. (13) results in the following residual error: ∂p~ þ c þ 2ð1 r Þx2 δðx1 Þ p~ ∂x1 ∂p~ þ cx2 þ kx1 þ μx31 þ ð1 r Þx2 jx2 jδðx1 Þ ∂x2
Δðx1 ; x2 ; aÞ ¼ x2
þ πK 1 ¼
x2
∂2 p~ ∂2 p~ þ πK 2 ɛ 2 x21 2 2 ∂x2 ∂x2
∂Q n þ c þ 2ð1 r Þx2 δðx1 Þ ∂x1
∂Q n þ cx2 þ kx1 þμx31 þ ð1 r Þx2 jx2 jδðx1 Þ ∂x2 "
#) 2 2 ∂Q n ∂ Qn þ πK 1 þ πK 2 ɛ 2 x21 þ p~ ðx1 ; x2 ; aÞ ∂x2 ∂x22 ~ 1 ; x2 ; aÞ ¼ Fðx1 ; x2 ; aÞpðx
(18)
~ 1 ; x2 ; aÞ to satisfy Eq. (13) is Fðx1 ; x2 ; aÞ ¼ 0 considering pðx ~ 1 ; x2 ; aÞ a 0. In the end of Eq. (18), the only possibility for pðx ~ 1 ; x2 ; aÞ is only an approximate solution. In these cases, another set of However, Fðx1 ; x2 ; aÞ a0 in most cases because pðx mutually independent functions H s ðx1 ; x2 Þ that span space RNp are introduced to make the projection of Fðx1 ; x2 ; aÞ on RNp vanish, which leads to Z þ1 Z þ1 Fðx1 ; x2 ; aÞH s ðx1 ; x2 Þ dx1 dx2 ¼ 0 (19) 1
1
~ 1 ; x2 ; aÞ in the weak sense of integration if Fðx1 ; x2 ; aÞ This means that the reduced FP equation (13) is fulfilled with pðx H s ðx1 ; x2 Þ is integrable in RNp . Selecting H s ðx1 ; x2 Þ as H s ðx1 ; x2 Þ ¼ xk1 l xl2 f 1 ðx1 Þf 2 ðx2 Þ
(20)
s ¼ 12 ðk þ 2Þðk 1Þ þ l þ 1;
Np nonlinear algebraic equations in terms of Np unknown where k ¼ 1; 2; …; n; l ¼ 0; 1; 2; …; k and parameters can be formulated. The algebraic equations can be solved with any available method to determine the parameters. Numerical experience shows that a convenient and effective choice for f 1 ðx1 Þ and f 2 ðx2 Þ is the PDFs obtained with the EQL method or Gaussian closure method as follows: ( ) x2 1 f 1 ðx1 Þ ¼ pffiffiffiffiffiffi exp 12 (21) 2s1 2π s1 ( ) x22 1 f 2 ðx2 Þ ¼ pffiffiffiffiffiffi exp 2 2s2 2π s2
(22)
Because of the particular choice for f 1 ðx1 Þ and f 2 ðx2 Þ, the integration in Eq. (19) can be easily evaluated in closed forms according to the properties of a Gaussian PDF. ~ 1 ; x2 ; aÞ is obtained for pðz; ~ z_ Þ, the PDFs of y and y_ (denoted as p~ Y ðyÞ and p~ Y_ ðyÞ, _ respectively) can be further When pðx achieved according to Eq. (5). The procedure follows the methodology for seeking the PDF distribution of a function of a random variable, which is introduced in the monograph of Lutes and Sarkani [27]. Herein, the procedure is briefly given. For the PDF of y, i.e., p~ Y ðyÞ, y is a function of z due to the relationship that y ¼ jzj in Eq. (5). It can be simply expressed as y ¼ gðzÞ and gðÞ is a general function. Therefore, p~ Z ½g j 1 ðyÞ p~ Y ðyÞ ¼ ∑ dgðuÞ j du 1 u ¼ gj
ðyÞ
(23)
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H.T. Zhu / Journal of Sound and Vibration 333 (2014) 954–961
with the summation being over all inverse points z ¼ g j 1 ðyÞ that map from z to y. g 1 ðÞ is the inverse function of gðÞ; R þ1 ~ z_ Þ dz_ . g j ðÞ can be formulated as follows: p~ Z ðzÞ ¼ 1 pðz; 8 z40 > < z; z¼0 y ¼ jzj ¼ gðzÞ ¼ 0; (24) > : z; z o 0 According to Eqs. (23) and (24), p~ Y ðyÞ ¼ p~ Zþ ðyÞ þ p~ Z ð yÞ; p~ Zþ ðÞ
y40
(25)
p~ Z ðÞ
and are the PDFs locating at the positive domain and the negative domain of z, respectively. Furthermore, where it is defined that p~ Y ð0Þ ¼ 2p~ Z ð0Þ considering Eq. (23) is null for z ¼0. _ i.e., p~ Y_ ðyÞ, _ y_ is a function of multiple random variables due to the relationship that y_ ¼ z_ sgnðzÞ in Eq. (5). For the PDF of y, _ can be evaluated using its cumulative Similarly, it is defined that y_ ¼ hðz; z_ Þ and hðÞ is also a general function. p~ Y_ ðyÞ distribution function Z Z _ ¼ ~ z_ Þ dz dz_ pðz; (26) F Y_ ðyÞ _ r y_ hðz;zÞ
In terms of hðz; z_ Þ ¼ z_ sgnðzÞ _ ¼ F Y_ ðyÞ
Z
Z
þ1
y_
dz 1
0
~ z_ Þ dz_ þ pðz;
Z
Z
0
dz 1
þ1 y_
~ z_ Þ dz_ pðz;
Taking the derivative with respect to y_ on Eq. (27) gives the probability density function as Z þ1 Z 0 _ ¼ ~ yÞ _ dz þ ~ yÞ _ dz pðz; pðz; p~ Y_ ðyÞ 0
(27)
(28)
1
3. Numerical analysis In the numerical analysis, the study considers a vibro-impact Duffing oscillator under a unilateral zero-offset barrier. Eqs. (1) and (2) give the equations of motion. The parameters of the system in these two equations are listed as follows: c ¼0.1, k ¼1, 2πK 1 ¼ 0:5, 2πK 2 ¼ 0:5 and r¼ 0.90. This study considers the case that the restitution coefficient is not very close to unity. The effects of the nonlinearity coefficient μ and the excitation factor ɛ are examined in the following analysis. A Monte Carlo simulation is also conducted to provide an adequate PDF for comparison. The number of samples is 1 107 for simulation.
3.1. Case 1: Weak nonlinearity in displacement In the first case, weak nonlinearity in displacement is considered in Eq. (1) as μ ¼ 0:1. The excitation factor is set as ɛ ¼ 0:1. Fig. 1a shows a comparison on the obtained PDF solutions using each method. EPC (n¼2) denotes the result given by the EPC method with the polynomial order being 2. The result shows that EPC (n ¼2) equals the one given by the equivalent linearization method. Fig. 1a shows there is a significant difference between the simulated result (MCS) and EPC (n ¼2). Furthermore, the logarithmic PDFs of displacement are compared in Fig. 1b showing more significant difference in the tail region between MCS and EPC (n ¼2). Because EPC (n ¼2) equals the one given by the equivalent linearization method, EPC (n¼2) denotes a Gaussian PDF. The comparison indicates that the PDF given by the equivalent linearization method is inadequate even in the case of weak nonlinearity. On the other hand, EPC (n¼4) gives satisfactory agreement with the simulated result as shown in Fig. 1a and b. In the case of velocity, the results are close to each other as shown in Fig. 1c and d. Furthermore, the simulated PDF of velocity shows a little asymmetric distribution due to the impacts with a small restitution coefficient (i.e., r ¼0.90). It is worth noting that a little difference exists between MCS and EPC (n ¼4) in the tail regions of PDFs. The formulation of the difference is due to the weak approximation on the response of the vibro-impact oscillators using a continuous Markov process. As Section 1 describes, the coefficient r is assumed to be close to unity. Therefore, (1 r) can be regarded as a small parameter. In such a case, the response of nonlinear systems may be much less significant discontinuous in its timederivative. The response of vibro-impact oscillators can be approximated with a continuous Markov process. Consequently the PDF of the response can be obtained by solving the corresponding FP equation. In this paper, the restitution coefficient of impacts is taken as 0.90 which is not very close to unity. Therefore, the use of a continuous Markov process is a little different from the actual response of vibro-impact oscillators. This leads the PDF of the FP equation to differ a little from the simulated result. This situation is also observed in the following case studies.
H.T. Zhu / Journal of Sound and Vibration 333 (2014) 954–961
0.8
0
EPC n=2 EPC n=4 MCS
0.7 0.6
−1 Log(PDF)
PDF
0.4 0.3
−2 −2.5 −3
0.1
−3.5 1
2
3
−4
4
0.35
0
0.3
−0.5
Log(PDF)
0.2 0.15 0.1
1
2
3
4
−4
−3
−2
−1
0
1
−1.5 −2 −2.5 −3
EPC n=2 EPC n=4 MCS
0.05 0 −5
0
−1
0.25 PDF
−1.5
0.2
0
EPC n=2 EPC n=4 MCS
−0.5
0.5
0
959
EPC n=2 EPC n=4 MCS
−3.5 2
3
4
5
−4 −5
−4
−3
−2
−1
0
1
2
3
4
5
Fig. 1. Comparison of PDFs in Case 1 (μ ¼ 0:1, ɛ ¼ 0:1): (a) PDFs of displacement; (b) logarithmic PDFs of displacement; (c) PDFs of velocity; (d) logarithmic PDFs of velocity.
3.2. Case 2: Strong nonlinearity in displacement Next, the nonlinearity coefficient increases from 0.1 to 1. As well known, the problem of strong nonlinearity is difficult to be solved in some methods. Figs. 2a and b shows a comparison on the PDFs obtained with each method for displacement in this case. EPC (n ¼2) further departs from the simulated result for the whole PDF and the tail region. Whereas, EPC (n¼4) still agrees well with MCS even in the tail region. Fig. 2c and d shows the results given by each method for velocity. EPC (n ¼2), EPC (n¼4) and MCS are close to each other. Because EPC (n ¼2) equals the one given by the equivalent linearization method and the result of equivalent linearization method is Gaussian, the simulated PDF of velocity shows an asymmetric shape which differs a little from being Gaussian. As expected, a little difference is observed between MCS and EPC (n ¼4) in the tail regions of PDFs. This is due to the difficulty of modeling the actual response of vibro-impact oscillators with a continuous Markov process when the restitution coefficient of impacts is not very close to unity (e.g., r ¼0.90).
3.3. Case 3: High-level parametric excitation in displacement Last, the intensity level of the parametric excitation increases as the excitation factor ɛ increases from 0.1 to 0.4 in this case. As shown in from Fig. 3a through d, similar conclusions to those of the above cases can be drawn. EPC (n¼2) differs a lot from the simulated result in the case of displacement. Comparatively, EPC (n ¼4) provides good approximate PDFs with MCS, especially in the tail region of the obtained PDFs of displacement. In the case of velocity, the results given with each method are close to each other. The simulated PDF of velocity shows an asymmetric shape which is a slight non-Gaussian distribution. Similarly, a little difference is observed between MCS and EPC (n ¼4) in the tail regions of PDFs. The reason has been discussed in the previous case studies. When the restitution coefficient of impacts is not very close to unity (e.g., r¼ 0.90 in this paper), the approximation of a Markov process is a little different from modeling the actual response of vibro-impact oscillators, which results in the difference of the PDF of the FP equation from the simulated result. The effects of the increase of the parametric excitation intensity can be observed in the tail regions of the PDFs of displacement and velocity as compared in Figs. 2b and 3b, and in Figs. 2d and 3d, respectively. This indicates that the parametric excitation has an effect in the tail regions of the PDFs of displacement and velocity.
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H.T. Zhu / Journal of Sound and Vibration 333 (2014) 954–961
1.2
0
EPC n=2 EPC n=4 MCS
1
−1 Log(PDF)
0.8 PDF
EPC n=2 EPC n=4 MCS
−0.5
0.6 0.4
−1.5 −2 −2.5 −3
0.2
−3.5 0
0.5
1
1.5
2
2.5
−4
3
0.4
0
0.35
−0.5
0.3
−1
0.25
−1.5
Log(PDF)
PDF
0
0.2 0.15 0.1
0 −5
−4
−3
−2
−1
0
0.5
1
1.5
2
3
4
−4 −5
5
2.5
3
−2
EPC n=2 EPC n=4 MCS
−3.5
1
2
−2.5 −3
EPC n=2 EPC n=4 MCS
0.05
0
−4
−3
−2
−1
0
1
2
3
4
5
Fig. 2. Comparison of PDFs in Case 2 (μ ¼ 1, ɛ ¼ 0:1): (a) PDFs of displacement; (b) logarithmic PDFs of displacement; (c) PDFs of velocity; (d) logarithmic PDFs of velocity. 1.2
0
EPC n=2 EPC n=4 MCS
1
−1 Log(PDF)
0.8 PDF
EPC n=2 EPC n=4 MCS
−0.5
0.6 0.4
−1.5 −2 −2.5 −3
0.2 0
−3.5 0
0.5
1
1.5
2
2.5
−4
3
0.35
0
0.3
−0.5
Log(PDF)
PDF
0.5
1
1.5
2
2.5
3
−1
0.25 0.2 0.15 0.1
−4
−3
−2
−1
0
1
−1.5 −2 −2.5 −3
EPC n=2 EPC n=4 MCS
0.05 0 −5
0
EPC n=2 EPC n=4 MCS
−3.5 2
3
4
5
−4 −5
−4
−3
−2
−1
0
1
2
3
4
5
Fig. 3. Comparison of PDFs in Case 3 (μ ¼ 1, ɛ ¼ 0:4): (a) PDFs of displacement; (b) logarithmic PDFs of displacement; (c) PDFs of velocity; (d) logarithmic PDFs of velocity.
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4. Conclusions This paper presents a solution procedure for the stationary PDF of vibro-impact Duffing oscillators under external and parametric Gaussian white noises. The procedure first adopts the Zhuravlev non-smooth coordinate transformation to convert the original vibro-impact oscillator into an oscillator without barriers. After that, the EPC method is proposed to solve the FP equation which governs the PDF of the response of the converted oscillator. In the numerical analysis, the study considers vibro-impact Duffing oscillators under external and parametric Gaussian white noises. The different levels of nonlinearity in displacement and parametric excitation intensity are examined to show the effectiveness of the proposed method. Comparison with the simulated results shows that the proposed solution procedure can present good approximate PDFs for both displacement and velocity. A little difference exists between MCS and EPC (n ¼4) in the tail regions of PDFs. Because the restitution coefficient of impacts is taken as 0.90 in this paper, the approximation of a continuous Markov process is a little different from modeling the actual response of vibro-impact oscillators. Consequently the approximate PDF of the FP equation differs a little from the simulated result. Acknowledgments This research is jointly supported by the National Basic Research Program of China (973 Program) under Grant no. 2013CB035904, the National Natural Science Foundation of China under Grant nos. 51008211, 51178308, and the Innovation Foundation of Tianjin University under Grant no. 60301014. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
R.A. Ibrahim, Vibro-Impact Dynamics: Modeling, Mapping and Applications, Springer, 2009. P.J. Holmes, The dynamics of repeated impacts with a sinusoidally vibrating table, Journal of Sound and Vibration 84 (2) (1982) 173–189. V.N. Pilipchuk, Impact modes in discrete vibrating systems with rigid barriers, International Journal of Non-Linear Mechanics 36 (6) (2001) 999–1012. S.W. Shaw, P.J. Holmes, A periodically forced impact oscillator with large dissipation, Journal of Applied Mechanics 50 (4a) (1983) 849–857. G.W. Luo, J.H. Xie, Hopf bifurcation of a two-degree-of-freedom vibro-impact system, Journal of Sound and Vibration 213 (3) (1998) 391–408. A.B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration 145 (2) (1991) 279–297. W. Chin, E. Ott, H.E. Nusse, C. Grebogi, Grazing bifurcations in impact oscillators, Physical Review E 50 (6) (1994) 4427–4444. J. de Weger, W. van de Water, J. Molenaar, Grazing impact oscillations, Physical Review E 62 (2) (2000) 2030–2041. M.F. Dimentberg, D.V. Iourtchenko, Random vibrations with impacts: a review, Nonlinear Dynamics 36 (2–4) (2004) 229–254. A. Baratta, Dynamics of a single-degree-of-freedom system with a unilateral obstacle, Structural Safety 8 (1–4) (1990) 181–194. H.G. Davies, Random vibration of a beam impacting stops, Journal of Sound and Vibration 68 (4) (1980) 479–487. Z.L. Huang, Z.H. Liu, W.Q. Zhu, Stationary response of multi-degree-of-freedom vibro-impact systems under white noise excitations, Journal of Sound and Vibration 275 (1–2) (2004) 223–240. N. Sri Namachchivaya, J.H. Park, Stochastic dynamics of impact oscillators, Journal of Applied Mechanics 72 (6) (2005) 862–870. J.Q. Feng, W. Xu, R. Wang, Stochastic responses of vibro-impact duffing oscillator excited by additive Gaussian noise, Journal of Sound and Vibration 309 (3–5) (2008) 730–738. J.Q. Feng, W. Xu, H.W. Rong, R. Wang, Stochastic responses of Duffing–Van der Pol vibro-impact system under additive and multiplicative random excitations, International Journal of Non-Linear Mechanics 44 (1) (2009) 51–57. D.V. Iourtchenko, L.L. Song, Numerical investigation of a response probability density function of stochastic vibroimpact systems with inelastic impacts, International Journal of Non-Linear Mechanics 41 (3) (2006) 447–455. I. Kovacic, M.J. Brennan, The Duffing Equation: Nonlinear Oscillators and their Behaviour, John Wiley & Sons Ltd., West Sussex, United Kingdom, 2011. P.L. Green, K. Worden, K. Atallah, N.D. Sims, The benefits of Duffing-type nonlinearities and electrical optimisation of a mono-stable energy harvester under white Gaussian excitations, Journal of Sound and Vibration 331 (20) (2012) 4504–4517. P.L. Green, K. Worden, N.D. Sims, On the identification and modelling of friction in a randomly excited energy harvester, Journal of Sound and Vibration 332 (19) (2013) 4696–4708. M. Vasta, A. Luongo, Dynamic analysis of linear and nonlinear oscillations of a beam under axial and transversal random Poisson pulses, Nonlinear Dynamics 36 (2–4) (2004) 421–435. H.T. Zhu, G.K. Er, V.P. Iu, K.P. Kou, Responses of nonlinear oscillators excited by nonzero-mean parametric Poisson impulses on displacement, Journal of Engineering Mechanics 138 (5) (2012) 450–457. V.F. Zhuravlev, A method for analyzing vibration-impact systems by means of special functions, Mechanics of Solids 11 (2) (1976) 23–27. G.K. Er, An improved closure method for analysis of nonlinear stochastic systems, Nonlinear Dynamics 17 (3) (1998) 285–297. G.K. Er, V.P. Iu, Stochastic response of based-excited Coulomb oscillator, Journal of Sound and Vibration 233 (1) (2000) 81–92. H.T. Zhu, G.K. Er, V.P. Iu, K.P. Kou, Probabilistic solution of nonlinear oscillators excited by combined Gaussian and Poisson white noises, Journal of Sound and Vibration 330 (12) (2011) 2900–2909. H.T. Zhu, Probabilistic solution of some multi-degree-of-freedom nonlinear systems under external independent Poisson white noises, Journal of the Acoustical Society of America 131 (6) (2012) 4550–4557. L.D. Lutes, S. Sarkani, Random Vibrations: Analysis of Structural and Mechanical Systems, Elsevier, 2004.
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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises H.T. Zhu n State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, People's Republic of China
a r t i c l e in f o
abstract
Article history: Received 26 August 2013 Accepted 2 October 2013 Handling Editor: K. Worden Available online 1 November 2013
This study presents a solution procedure for the stationary probability density function (PDF) of the response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises. First the Zhuravlev non-smooth coordinate transformation is adopted to convert a vibro-impact oscillator into an oscillator without barriers. The stationary PDF of the converted oscillator is governed by the Fokker–Planck (FP) equation. The FP equation is solved by the exponential-polynomial closure (EPC) method. Illustrative examples are presented with vibro-impact Duffing oscillators under external and parametric Gaussian white noises to show the effectiveness of the solution procedure. The parametric excitation is acting in displacement and the constraint is a unilateral zerooffset barrier. The restitution coefficient of impacts is taken as 0.90. Comparison with the simulated results shows that the proposed solution procedure can provide good approximate PDFs for displacement and velocity although a little difference exists in the tail of these PDFs. This difference may be due to the weak approximation on the response of the vibro-impact oscillators using a continuous Markov process when the restitution coefficient is not very close to unity. & 2013 Elsevier Ltd. All rights reserved.
1. Introduction The stochastic response of structures or systems under random excitation is a critical issue in reliability analysis. An overlarge response may be a great threat to the safety of life and property. Therefore, many efforts are made on the evaluation of the stochastic response of these systems. In particular, vibro-impact systems have attracted much attention in the scientific community because the systems can model many relevant problems in the field of science and engineering. For example, vibro-impact systems can describe ship roll motion against icebergs, rotor–stator rubbing in rotating machinery or impact interactions in pipe–baffle interfaces due to seismic excitations, etc. The modeling, mapping and applications of vibro-impact systems are extensively introduced by Ibrahim [1]. The behavior of vibro-impact systems has been widely studied in the case of deterministic or random excitations. In the case of deterministic excitation, many interesting dynamic phenomena were observed, e.g., subharmonic oscillation [2], impact modes [3], chaotic motion [4,5], grazing bifurcation [6–8]. In the case of random excitation, many researches have also been conducted and reported. Dimentberg and Iourtchenko presented a comprehensive review on the investigation on vibro-impact systems under stochastic excitations [9]. In those investigations, the stochastic response of vibro-impact systems was evaluated by statistical moments or probability density functions [10]. An equivalent linearization method was used for analyzing
n
Tel.: þ86 22 27400843; fax: þ86 22 27404319. E-mail address: [email protected]
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random vibration of a beam impacting stops [11]. This method is limited to the case of light nonlinearity and nearly elastic impact. A stochastic averaging method is another approach to deal with the problem of vibro-impact systems [12–15]. The stochastic averaging method is suitable in the case of lightly damping and weak wide-band excitation. Monte Carlo simulation is a versatile technique [16]. However, extensive efforts are needed to simulate impacts. The procedure of impact tracing also introduces additional points when simulating each impact. So, in order to simulate an adequate PDF, both response interval selection and sample selection around the barrier are critical in this simulation technique. Besides, when the tail of response PDF is considered, the simulation time becomes tremendous. On the other hand, the Duffing oscillator in its various forms has been adopted to model many nonlinear physical systems, such as large deflection of a beam with nonlinear stiffness, vibration of a beam with nonlinear stiffness due to inplane tension, nonlinear cable vibrations and nonlinear electrical circuit [17]. Furthermore, consideration of Duffing-type nonlinearities in some mechanical systems can lead these mechanical systems to improve their performance. For example, introduction of Duffing-type nonlinearities into a mono-stable energy harvester can reduce the maximum displacement amplitude of the center magnet without effecting device performance, which can be used to create smaller energy harvesting devices [18]. In this research field, most previous researches were devoted to the case of sinusoidal excitation. The case of ambient vibration has been becoming a popular research topic recently because an energy harvesting device sometimes performs ambient vibration which is stochastic in nature [19]. When ambient vibration is considered, the excitation is usually treated as Gaussian white noise and the response of this device can be evaluated using the Fokker– Planck (FP) equation. Therefore, an investigation on the Duffing oscillator in its various forms under different types of excitations is fundamental for further investigating many nonlinear physical systems. This paper presents a study on the PDF of the stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises. The constraint is a unilateral zero-offset barrier. The vibro-impact Duffing oscillator can be used to model random vibration of a beam with a unilateral barrier under axial and transversal random excitations [20,21]. In particular, the probability density function (PDF) and the tail PDF of these Duffing oscillators are formulated, which is less addressed in previous studies. In this solution procedure, the Zhuravlev non-smooth coordinate transformation is first introduced to the equation of motion so that the transformed equation does not contain any impact terms [1,22]. The PDF of the response of the transformed oscillator is governed by the FP equation which is solved by the exponentialpolynomial closure (EPC) method [23–26]. The cases of light nonlinearity and strong nonlinearity are further investigated as well as the intensity level of parametric excitation. In these analyses, the restitution coefficient is 0.90, which is not very close to unity. Comparison with the simulated results shows that the proposed solution procedure can provide good approximate PDFs for displacement and velocity although a little difference exists in the tail of these PDFs. The formulation of this difference may be due to weakly modeling the response of the vibro-impact oscillators using a continuous Markov process when the restitution coefficient is not very close to unity. 2. Problem formulation A vibro-impact Duffing oscillator under a unilateral zero-offset barrier can be formulated by y€ þcy_ þ ky þ μy3 ¼ ξ1 ðtÞ þɛyξ2 ðtÞ; y_ þ ¼ r y_ ;
y 40
y ¼ 0; 0 o r r1
(1) (2)
€ y; _ y are the acceleration, velocity and displacement of the oscillator, respectively; c is the damping coefficient; k is where y; the linear stiffness coefficient; μ is the nonlinearity coefficient and ξi ðtÞ are the zero-mean mutually independent Gaussian white noises. The vibro-impact Duffing oscillator can be used to model random vibration of a beam with a unilateral barrier under axial and transversal random excitations [20,21]. The mean and correlation function of ξi ðtÞ are given below E½ξi ðtÞ ¼ 0
(3)
E½ξi ðtÞξi ðt þ τÞ ¼ 2πK i δðτÞ
(4)
where E½ is the expectation; 2πK i is the excitation intensity of ξi ðtÞ and δðÞ is the Dirac delta function. The use of the Zhuravlev non-smooth coordinate transformation leads the above vibro-impact oscillator to be converted into an oscillator without barriers. The detailed procedure was presented in either the monograph of Ibrahim [1] or the review paper written by Dimentberg and Iourtchenko [9]. The transformation procedure is briefly presented herein y ¼ jzj;
y_ ¼ z_ sgnðzÞ;
y€ ¼ z€ sgnðzÞ
(5)
where z€ ; z_ ; z are the acceleration, velocity and displacement of the converted oscillator, respectively, and sgnðÞ is the sign function. Eq. (5) is formulated because y ¼ jzj is equivalent to y ¼ z sgnðzÞ and z½dðsgnðzÞÞ=dt ¼ 0 noting that sgn(z) changes its sign at z¼0. Eq. (2) gives the condition for the case of inelastic impacts. In the transformation procedure, the coefficient r is assumed to be close to unity. Therefore, (1 r) can be regarded as a small parameter. In such a case, the response of nonlinear systems may be much less significant discontinuous in its time-derivative. This allows some conventional approximation techniques
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solving nonlinear stochastic differential equations to be used [10,14]. For example, the response of vibro-impact oscillators can be approximated with a continuous Markov process. Consequently the PDF of the response can be obtained by solving the corresponding FP equation. According to Eq. (5), Eq. (2) is converted to z_ þ ¼ r z_
at z ¼ 0
(6)
where z_ ; z_ þ represent the impact velocities before and after an impact for the converted oscillator, respectively. Therefore, the converted velocity jump is reduced by an amount proportional to (1 r). It is reasonable to use the Dirac delta-function to introduce this jump into the equation of motion as an additional impulsive damping term. The additional damping term due to impacts may be written in the form: ðz_ þ z_ Þδðt t i Þ ¼ ð1 rÞz_ δðt t i Þ;
given that jz_ þ j o jz_ jo jz_ j
(7)
where ti is the time instant of impacts (i.e., zðt i Þ ¼ 0). Correspondingly, the variables are converted from the time domain to the space domain. This can be fulfilled by setting ðt t i Þ ¼ zðtÞ=z_ ðt i Þ. It is because zðtÞ ¼ zðt i Þ þ z_ ðt i Þðt t i Þ in a small interval and zðt i Þ ¼ 0 at an impact. Therefore, δðt t i Þ ¼ δðzðtÞ=z_ ðt i ÞÞ and δðt t i Þ ¼ jz_ jδðzÞ. In this case, Eq. (8) can be written in the form: ð1 rÞz_ δðt t i Þ ¼ ð1 rÞz_ jz_ jδðzÞ
(8)
After the transformation process, Eqs. (1) and (2) are combined into one equation by adding the above impulsive damping term: z€ þ cz_ þ kzþμz3 þ ð1 rÞz_ jz_ jδðzÞ ¼ sgnðzÞξ1 ðtÞ þɛzξ2 ðtÞ Let x1 ¼ z, x2 ¼ z_ , Eq. (9) can be expressed as ( x_ 1 ¼ x2 x_ 2 ¼ cx2 kx1 μx31 ð1 rÞx2 jx2 jδðx1 Þ þ sgnðx1 Þξ1 ðtÞ þ ɛx1 ξ2 ðtÞ
(9)
(10)
The response vector fx1 ; x2 gT is Markovian and the PDFs of responses are governed by the following FP equation: ∂p ∂p ∂ ¼ x2 þ cx2 þkx1 þ μx31 þ ð1 r Þx2 x2 δðx1 Þ p ∂t ∂x1 ∂x2 þ
1 ∂2 p 1 ∂2 p 2πK 1 ðsgnðx1 ÞÞ2 2 þ 2πK 2 ɛ2 x21 2 2 ∂x2 2 ∂x2
(11)
In this paper, the stationary PDF solution is considered and the term on the left side of Eq. (11) is zero. Therefore, Eq. (11) is reduced as follows: x2
∂p þ c þ ð1 r Þx2 δðx1 Þ þ ð1 r Þx2 sgnðx2 Þδðx1 Þ p ∂x1 ∂p þ cx2 þkx1 þ μx31 þ ð1 r Þx2 jx2 jδðx1 Þ ∂x2 þ πK 1
∂2 p ∂2 p þ πK 2 ɛ 2 x21 2 ¼ 0 2 ∂x2 ∂x2
(12)
Considering jx2 j ¼ x2 sgnðx2 Þ, Eq. (12) finally reads x2
∂p þ c þ 2ð1 r Þx2 δðx1 Þ p ∂x1
∂p þ cx2 þkx1 þ μx31 þ ð1 r Þx2 jx2 jδðx1 Þ ∂x2 þ πK 1
∂2 p ∂2 p þ πK 2 ɛ 2 x21 2 ¼ 0 2 ∂x2 ∂x2
(13)
In general the following requirements are satisfied by the PDF pðx1 ; x2 Þ of the stationary response of the nonlinear oscillator: 8 > pðx1 ; x2 Þ Z 0; x1 ; x2 A R2 > > < lim pðx1 ; x2 Þ ¼ 0; i ¼ 1; 2 (14) xi - 7 1 > > R > þ1 R þ1 : 1 1 pðx1 ; x2 Þ dx1 dx2 ¼ 1 where R denotes real space. The exact PDF solution of Eq. (13) is usually not obtainable. According to the above requirements, an approximate PDF is proposed and it is expressed as an exponential-polynomial function of state variables. ~ 1 ; x2 ; aÞ to Eq. (13) is assumed to be [23–26] The approximate PDF solution pðx ~ 1 ; x2 ; aÞ ¼ C eQ n ðx1 ;x2 ;aÞ pðx
(15)
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where C is a normalization constant and a is an unknown parameter vector containing Np entries. The polynomial Q n ðx1 ; x2 ; aÞ is expressed as n
i
Q n ðx1 ; x2 ; aÞ ¼ ∑ ∑ aij xi1 j xj2
(16)
i¼1j¼0
which is an nth-degree polynomial in x1 and x2. To fulfill the requirements (14), it is also required that lim Q n ðx1 ; x2 ; aÞ ¼ 1;
i ¼ 1; 2:
xi - 7 1
(17)
~ 1 ; x2 ; aÞ is only an approximation of pðx1 ; x2 Þ and the number of unknown parameters Np is always limited in In general, pðx ~ 1 ; x2 ; aÞ practice. Therefore, the FP equation (13) cannot be satisfied exactly with the approximate solution. Substituting pðx into Eq. (13) results in the following residual error: ∂p~ þ c þ 2ð1 r Þx2 δðx1 Þ p~ ∂x1 ∂p~ þ cx2 þ kx1 þ μx31 þ ð1 r Þx2 jx2 jδðx1 Þ ∂x2
Δðx1 ; x2 ; aÞ ¼ x2
þ πK 1 ¼
x2
∂2 p~ ∂2 p~ þ πK 2 ɛ 2 x21 2 2 ∂x2 ∂x2
∂Q n þ c þ 2ð1 r Þx2 δðx1 Þ ∂x1
∂Q n þ cx2 þ kx1 þμx31 þ ð1 r Þx2 jx2 jδðx1 Þ ∂x2 "
#) 2 2 ∂Q n ∂ Qn þ πK 1 þ πK 2 ɛ 2 x21 þ p~ ðx1 ; x2 ; aÞ ∂x2 ∂x22 ~ 1 ; x2 ; aÞ ¼ Fðx1 ; x2 ; aÞpðx
(18)
~ 1 ; x2 ; aÞ to satisfy Eq. (13) is Fðx1 ; x2 ; aÞ ¼ 0 considering pðx ~ 1 ; x2 ; aÞ a 0. In the end of Eq. (18), the only possibility for pðx ~ 1 ; x2 ; aÞ is only an approximate solution. In these cases, another set of However, Fðx1 ; x2 ; aÞ a0 in most cases because pðx mutually independent functions H s ðx1 ; x2 Þ that span space RNp are introduced to make the projection of Fðx1 ; x2 ; aÞ on RNp vanish, which leads to Z þ1 Z þ1 Fðx1 ; x2 ; aÞH s ðx1 ; x2 Þ dx1 dx2 ¼ 0 (19) 1
1
~ 1 ; x2 ; aÞ in the weak sense of integration if Fðx1 ; x2 ; aÞ This means that the reduced FP equation (13) is fulfilled with pðx H s ðx1 ; x2 Þ is integrable in RNp . Selecting H s ðx1 ; x2 Þ as H s ðx1 ; x2 Þ ¼ xk1 l xl2 f 1 ðx1 Þf 2 ðx2 Þ
(20)
s ¼ 12 ðk þ 2Þðk 1Þ þ l þ 1;
Np nonlinear algebraic equations in terms of Np unknown where k ¼ 1; 2; …; n; l ¼ 0; 1; 2; …; k and parameters can be formulated. The algebraic equations can be solved with any available method to determine the parameters. Numerical experience shows that a convenient and effective choice for f 1 ðx1 Þ and f 2 ðx2 Þ is the PDFs obtained with the EQL method or Gaussian closure method as follows: ( ) x2 1 f 1 ðx1 Þ ¼ pffiffiffiffiffiffi exp 12 (21) 2s1 2π s1 ( ) x22 1 f 2 ðx2 Þ ¼ pffiffiffiffiffiffi exp 2 2s2 2π s2
(22)
Because of the particular choice for f 1 ðx1 Þ and f 2 ðx2 Þ, the integration in Eq. (19) can be easily evaluated in closed forms according to the properties of a Gaussian PDF. ~ 1 ; x2 ; aÞ is obtained for pðz; ~ z_ Þ, the PDFs of y and y_ (denoted as p~ Y ðyÞ and p~ Y_ ðyÞ, _ respectively) can be further When pðx achieved according to Eq. (5). The procedure follows the methodology for seeking the PDF distribution of a function of a random variable, which is introduced in the monograph of Lutes and Sarkani [27]. Herein, the procedure is briefly given. For the PDF of y, i.e., p~ Y ðyÞ, y is a function of z due to the relationship that y ¼ jzj in Eq. (5). It can be simply expressed as y ¼ gðzÞ and gðÞ is a general function. Therefore, p~ Z ½g j 1 ðyÞ p~ Y ðyÞ ¼ ∑ dgðuÞ j du 1 u ¼ gj
ðyÞ
(23)
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with the summation being over all inverse points z ¼ g j 1 ðyÞ that map from z to y. g 1 ðÞ is the inverse function of gðÞ; R þ1 ~ z_ Þ dz_ . g j ðÞ can be formulated as follows: p~ Z ðzÞ ¼ 1 pðz; 8 z40 > < z; z¼0 y ¼ jzj ¼ gðzÞ ¼ 0; (24) > : z; z o 0 According to Eqs. (23) and (24), p~ Y ðyÞ ¼ p~ Zþ ðyÞ þ p~ Z ð yÞ; p~ Zþ ðÞ
y40
(25)
p~ Z ðÞ
and are the PDFs locating at the positive domain and the negative domain of z, respectively. Furthermore, where it is defined that p~ Y ð0Þ ¼ 2p~ Z ð0Þ considering Eq. (23) is null for z ¼0. _ i.e., p~ Y_ ðyÞ, _ y_ is a function of multiple random variables due to the relationship that y_ ¼ z_ sgnðzÞ in Eq. (5). For the PDF of y, _ can be evaluated using its cumulative Similarly, it is defined that y_ ¼ hðz; z_ Þ and hðÞ is also a general function. p~ Y_ ðyÞ distribution function Z Z _ ¼ ~ z_ Þ dz dz_ pðz; (26) F Y_ ðyÞ _ r y_ hðz;zÞ
In terms of hðz; z_ Þ ¼ z_ sgnðzÞ _ ¼ F Y_ ðyÞ
Z
Z
þ1
y_
dz 1
0
~ z_ Þ dz_ þ pðz;
Z
Z
0
dz 1
þ1 y_
~ z_ Þ dz_ pðz;
Taking the derivative with respect to y_ on Eq. (27) gives the probability density function as Z þ1 Z 0 _ ¼ ~ yÞ _ dz þ ~ yÞ _ dz pðz; pðz; p~ Y_ ðyÞ 0
(27)
(28)
1
3. Numerical analysis In the numerical analysis, the study considers a vibro-impact Duffing oscillator under a unilateral zero-offset barrier. Eqs. (1) and (2) give the equations of motion. The parameters of the system in these two equations are listed as follows: c ¼0.1, k ¼1, 2πK 1 ¼ 0:5, 2πK 2 ¼ 0:5 and r¼ 0.90. This study considers the case that the restitution coefficient is not very close to unity. The effects of the nonlinearity coefficient μ and the excitation factor ɛ are examined in the following analysis. A Monte Carlo simulation is also conducted to provide an adequate PDF for comparison. The number of samples is 1 107 for simulation.
3.1. Case 1: Weak nonlinearity in displacement In the first case, weak nonlinearity in displacement is considered in Eq. (1) as μ ¼ 0:1. The excitation factor is set as ɛ ¼ 0:1. Fig. 1a shows a comparison on the obtained PDF solutions using each method. EPC (n¼2) denotes the result given by the EPC method with the polynomial order being 2. The result shows that EPC (n ¼2) equals the one given by the equivalent linearization method. Fig. 1a shows there is a significant difference between the simulated result (MCS) and EPC (n ¼2). Furthermore, the logarithmic PDFs of displacement are compared in Fig. 1b showing more significant difference in the tail region between MCS and EPC (n ¼2). Because EPC (n ¼2) equals the one given by the equivalent linearization method, EPC (n¼2) denotes a Gaussian PDF. The comparison indicates that the PDF given by the equivalent linearization method is inadequate even in the case of weak nonlinearity. On the other hand, EPC (n¼4) gives satisfactory agreement with the simulated result as shown in Fig. 1a and b. In the case of velocity, the results are close to each other as shown in Fig. 1c and d. Furthermore, the simulated PDF of velocity shows a little asymmetric distribution due to the impacts with a small restitution coefficient (i.e., r ¼0.90). It is worth noting that a little difference exists between MCS and EPC (n ¼4) in the tail regions of PDFs. The formulation of the difference is due to the weak approximation on the response of the vibro-impact oscillators using a continuous Markov process. As Section 1 describes, the coefficient r is assumed to be close to unity. Therefore, (1 r) can be regarded as a small parameter. In such a case, the response of nonlinear systems may be much less significant discontinuous in its timederivative. The response of vibro-impact oscillators can be approximated with a continuous Markov process. Consequently the PDF of the response can be obtained by solving the corresponding FP equation. In this paper, the restitution coefficient of impacts is taken as 0.90 which is not very close to unity. Therefore, the use of a continuous Markov process is a little different from the actual response of vibro-impact oscillators. This leads the PDF of the FP equation to differ a little from the simulated result. This situation is also observed in the following case studies.
H.T. Zhu / Journal of Sound and Vibration 333 (2014) 954–961
0.8
0
EPC n=2 EPC n=4 MCS
0.7 0.6
−1 Log(PDF)
0.4 0.3
−2 −2.5 −3
0.1
−3.5 1
2
3
−4
4
0.35
0
0.3
−0.5
Log(PDF)
0.2 0.15 0.1
1
2
3
4
−4
−3
−2
−1
0
1
−1.5 −2 −2.5 −3
EPC n=2 EPC n=4 MCS
0.05 0 −5
0
−1
0.25 PDF
−1.5
0.2
0
EPC n=2 EPC n=4 MCS
−0.5
0.5
0
959
EPC n=2 EPC n=4 MCS
−3.5 2
3
4
5
−4 −5
−4
−3
−2
−1
0
1
2
3
4
5
Fig. 1. Comparison of PDFs in Case 1 (μ ¼ 0:1, ɛ ¼ 0:1): (a) PDFs of displacement; (b) logarithmic PDFs of displacement; (c) PDFs of velocity; (d) logarithmic PDFs of velocity.
3.2. Case 2: Strong nonlinearity in displacement Next, the nonlinearity coefficient increases from 0.1 to 1. As well known, the problem of strong nonlinearity is difficult to be solved in some methods. Figs. 2a and b shows a comparison on the PDFs obtained with each method for displacement in this case. EPC (n ¼2) further departs from the simulated result for the whole PDF and the tail region. Whereas, EPC (n¼4) still agrees well with MCS even in the tail region. Fig. 2c and d shows the results given by each method for velocity. EPC (n ¼2), EPC (n¼4) and MCS are close to each other. Because EPC (n ¼2) equals the one given by the equivalent linearization method and the result of equivalent linearization method is Gaussian, the simulated PDF of velocity shows an asymmetric shape which differs a little from being Gaussian. As expected, a little difference is observed between MCS and EPC (n ¼4) in the tail regions of PDFs. This is due to the difficulty of modeling the actual response of vibro-impact oscillators with a continuous Markov process when the restitution coefficient of impacts is not very close to unity (e.g., r ¼0.90).
3.3. Case 3: High-level parametric excitation in displacement Last, the intensity level of the parametric excitation increases as the excitation factor ɛ increases from 0.1 to 0.4 in this case. As shown in from Fig. 3a through d, similar conclusions to those of the above cases can be drawn. EPC (n¼2) differs a lot from the simulated result in the case of displacement. Comparatively, EPC (n ¼4) provides good approximate PDFs with MCS, especially in the tail region of the obtained PDFs of displacement. In the case of velocity, the results given with each method are close to each other. The simulated PDF of velocity shows an asymmetric shape which is a slight non-Gaussian distribution. Similarly, a little difference is observed between MCS and EPC (n ¼4) in the tail regions of PDFs. The reason has been discussed in the previous case studies. When the restitution coefficient of impacts is not very close to unity (e.g., r¼ 0.90 in this paper), the approximation of a Markov process is a little different from modeling the actual response of vibro-impact oscillators, which results in the difference of the PDF of the FP equation from the simulated result. The effects of the increase of the parametric excitation intensity can be observed in the tail regions of the PDFs of displacement and velocity as compared in Figs. 2b and 3b, and in Figs. 2d and 3d, respectively. This indicates that the parametric excitation has an effect in the tail regions of the PDFs of displacement and velocity.
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1
−1 Log(PDF)
0.8 PDF
EPC n=2 EPC n=4 MCS
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−1.5 −2 −2.5 −3
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3
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0.2 0.15 0.1
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−4
−3
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−1
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1
1.5
2
3
4
−4 −5
5
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EPC n=2 EPC n=4 MCS
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EPC n=2 EPC n=4 MCS
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−3
−2
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1
2
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Fig. 2. Comparison of PDFs in Case 2 (μ ¼ 1, ɛ ¼ 0:1): (a) PDFs of displacement; (b) logarithmic PDFs of displacement; (c) PDFs of velocity; (d) logarithmic PDFs of velocity. 1.2
0
EPC n=2 EPC n=4 MCS
1
−1 Log(PDF)
0.8 PDF
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0.6 0.4
−1.5 −2 −2.5 −3
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EPC n=2 EPC n=4 MCS
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3
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−4
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−2
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Fig. 3. Comparison of PDFs in Case 3 (μ ¼ 1, ɛ ¼ 0:4): (a) PDFs of displacement; (b) logarithmic PDFs of displacement; (c) PDFs of velocity; (d) logarithmic PDFs of velocity.
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4. Conclusions This paper presents a solution procedure for the stationary PDF of vibro-impact Duffing oscillators under external and parametric Gaussian white noises. The procedure first adopts the Zhuravlev non-smooth coordinate transformation to convert the original vibro-impact oscillator into an oscillator without barriers. After that, the EPC method is proposed to solve the FP equation which governs the PDF of the response of the converted oscillator. In the numerical analysis, the study considers vibro-impact Duffing oscillators under external and parametric Gaussian white noises. The different levels of nonlinearity in displacement and parametric excitation intensity are examined to show the effectiveness of the proposed method. Comparison with the simulated results shows that the proposed solution procedure can present good approximate PDFs for both displacement and velocity. A little difference exists between MCS and EPC (n ¼4) in the tail regions of PDFs. Because the restitution coefficient of impacts is taken as 0.90 in this paper, the approximation of a continuous Markov process is a little different from modeling the actual response of vibro-impact oscillators. Consequently the approximate PDF of the FP equation differs a little from the simulated result. Acknowledgments This research is jointly supported by the National Basic Research Program of China (973 Program) under Grant no. 2013CB035904, the National Natural Science Foundation of China under Grant nos. 51008211, 51178308, and the Innovation Foundation of Tianjin University under Grant no. 60301014. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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